Finite and infinite quotients of discrete and indiscrete groups

TL;DR

This paper explores the construction of simple groups via lattices in products of trees, emphasizing non-discrete groups and the methods for establishing finite and infinite quotients, with historical context and open problems.

Contribution

It introduces a novel approach to constructing simple groups using lattices in products of trees, highlighting the role of non-discrete groups and a two-step proof strategy.

Findings

Construction of simple groups from lattices in products of trees.

Distinct methods for proving the existence of finite and infinite quotients.

Discussion of the relation between hyperbolic groups and finite simple groups.

Abstract

These notes are devoted to lattices in products of trees and related topics. They provide an introduction to the construction, by M. Burger and S. Mozes, of examples of such lattices that are simple as abstract groups. Two features of that construction are emphasized: the relevance of non-discrete locally compact groups, and the two-step strategy in the proof of simplicity, addressing separately, and with completely different methods, the existence of finite and infinite quotients. A brief history of the quest for finitely generated and finitely presented infinite simple groups is also sketched. A comparison with Margulis' proof of Kneser's simplicity conjecture is discussed, and the relevance of the Classification of the Finite Simple Groups is pointed out. A final chapter is devoted to finite and infinite quotients of hyperbolic groups and their relation to the asymptotic properties of the finite simple groups. Numerous open problems are discussed along the way.